Mathematical Modeling for the Optimal Cost Design of Circular Isolated Footings with Eccentric Column

Abstract

This article shows a model for the design of circular isolated footings and the column placed anywhere in the footing under minimum cost criteria. Some designs for obtaining the diameter, effective depth, and steel areas of the footing under biaxial bending assume the maximum and uniform pressure at the bottom of the footing supported on elastic soils. All these works consider the column placed at the center of the footing. Three numerical problems are given (each problem presents four variants) to determine the lowest cost to design the circular footings under biaxial bending. Problem 1: Column without eccentricity. Problem 2: Column with eccentricity in the direction of the X axis of one quarter of the diameter of the footing. Problem 3: Column placed at the end furthest from the center of the footing on the X axis. The results are verified by the balance of moments, one-way shear or shear and two-way shear or punching. The new model shows a saving of 17.92% in the contact area with soil and of 31.15% in cost compared to the model proposed by other authors. In this way, the proposed minimum cost design model for circular footings will be of great help for the design when the column is placed on the center or edge of the footing.

1. Introduction

The essential function of the foundation elements is to transmit the loads of the structure to the subsoil. There are basic parameters to consider when choosing the type of foundation to be used; these elements are chosen mainly depending on the type of structure to be supported and the conditions and characteristics of support of the support soil. The foundations are divided into deep and shallow.

The main works of various researchers on foundation structures or structural footings in the last ten years are as follows: Zedan et al. [1] introduced three optimization techniques, such as Hooke and Jeeves, Fletcher–Reeves and Davidon–Fletcher–Powell, applied to the design of a circular footing adjacent to slopes. Basudhar and Dey [2] analyzed the optimal cost of circular footing design under biaxial bending using an unconstrained sequential minimization technique in conjunction with Powell’s conjugate direction method for multidimensional search and quadratic interpolation method for one dimensional minimization. Hassaan [3] proposed an optimal design using MATLAB for shallow foundations on sandy soils. Luévanos-Rojas [4] designed the circular isolated footings using a novel model. Momeni et al. [5] proposed a predictive model based in artificial neural networks to find the axial load capacity of piles and its distribution. Moghaddas-Tafreshi et al. [6] described a series of small-scale tests, at unit gravity, carried out on circular footings supported by reinforced sand. Luévanos-Rojas [7] investigated the comparison between the circular and rectangular isolated footings, and the circular footings presented more savings. Rezaei et al. [8] estimated the thin-walled shallow foundations load capacity: an experimental study by means of artificial intelligence. Khatri et al. [9] developed the behavior of the settlement and the pressure of the soil in skirted footings (square and rectangular) supported on sandy soil. Da Silva et al. [10] used statistics to predict the failure probability of pile foundations. Maheshwari [11] analyzed the combined footings that rest on a granular bed reinforced with extensible geosynthetic on improved ground of stone column. Munévar-Peña et al. [12] incorporated the random uncertainty of soil properties into the geostatistical study applied to the safe design of shallow foundations. Rawat and Mittal [13] studied the optimal design of footings with eccentric load. Rodrigo-Garcia and Rocha-de Albuquerque [14] developed a model for the nonlinear behavior applied to deep foundations to predict the settlement. Zhang et al. [15] investigated the bearing behavior of isolated footing substructures with reinforced concrete column. López-Chavarría et al. [16] designed the circular isolated footings under the lowest cost criterion assuming linear soil pressure. Liu and Jiang [17] proposed a method to study the deformation and consolidation characteristics of gentle rock and gentle soil foundation with hydrological moisture environment to solve this problem. Al-Abbas et al. [18] showed an experimental study of elastic deformation under an isolated footing. Gnananandarao et al. [19] applied multivariable regression analysis and artificial neural networks to predict the load-bearing capacity and settlement of multi-edge skirted footings on sand. Alelvan et al. [20] presented a comparison of problems applied to geotechnics, including large displacements and deformations using the Material Point Method and Finite Element Method associated with the Arbitrary Lagrangian Eulerian Method. Al-Ansari and Afzal [21] developed an analytical simplified model to design irregularly shaped footings that support a square column. López-Machado et al. [22] estimated a comparison for structural design of two reinforced concrete buildings of six levels taking into account the soil-structure interaction. Badakhshan and Noorzad [23] presented the results from laboratory modeling tests and numerical studies carried out on circular and square footings, assuming the same plan area that rests on geosynthetic reinforced sand bed. Fathipour et al. [24] proposed a numerical study to obtain the lateral soil pressure on geosynthetic-reinforced retaining structures by finite element and second-order cone programming. Solorzano and Plevris [25] presented the optimal design of rectangular footings using genetic algorithms according to the ACI 318-19 standard of the American Concrete Institute. Waheed et al. [26] developed and applied a parametric investigation for the optimal design of footings based on the Metaheuristic Practical Tool. Ekbote and Nainegali [27] studied two asymmetric footings interfering, taking into account the different widths and the effect of embedment depth to improve the ultimate load capacity and limit settlement within the working range. Komolafe et al. [28] investigated the square and circular footings supporting on cohesionless soils from a structural, geotechnical and construction cost perspective. Helis et al. [29] estimated the bearing capacity of a circular footing supported on sand using two reinforcing systems, the geogrid and the grid anchor.

According to the literature reviewed, the works closest to the topic addressed in this paper are as follows: Zedan et al. [1], Basudhar and Dey [2], López-Chavarría et al. [16] and Al-Ansari and Afzal [21] presented the minimum cost design for circular isolated footings supporting a concentric column (column located in the center of the footing). There are no circular isolated footings with eccentric columns; therefore, this document will be of great help to design when the columns are placed on the edge of the footing (property boundary).

This works shows a novel model to design circular isolated footings with minimum cost, assuming that the column is located anywhere on the footing, which is the main contribution. The proposed model is presented in two phases. Phase one consists of determining the minimum surface or area and the diameter, and phase two is finding the lowest cost once the diameter of the circular footing is known. Three numerical problems (each example presents four variants) are shown to design circular footings with minimum cost under biaxial bending. Problem 1: The center of the column is placed in the center of the footing. Problem 2: The center of the column is placed at a distance of 0.25 times the diameter of the footing from the center of the footing. Problem 3: One face of the column is placed on one of the faces of the footing.

2. Formulation of the Model

Figure 1 shows the distribution of soil pressure under the footing according to the type of soil and the stiffness of the footing [30].

Figure 2 shows a circular footing under biaxial bending, assuming that the column is placed anywhere on the footing supported on elastic soils and the soil pressure distribution is linear.

The general biaxial bending equation is:

$$\sigma =\frac{P}{A}+\frac{M_{xT}y}{I_x}+\frac{M_{yT}x}{I_y},$$

(1)

where: σ = stress provided by the soil at any location on the footing (kN/m²); P = unfactored vertical axial load (kN); A = surface or area of contact with soil at the bottom of the footing (m²); M_xT = unfactored total moment on the X axis (kN-m); M_yT = unfactored total moment on the Y axis (kN-m); I_x = moment of inertia on the X axis (m⁴); I_y = moment of inertia on the Y axis (m⁴); x = coordinate in the X direction of the base (m); y = coordinate in the Y direction of the base (m).

Equation (1) assumes that the soil contact surface on the footing operates completely in compression.

It can be considered a moment since the moment of inertia does not vary when the axes are rotated; therefore, a resultant moment M_R can be used [3,6,16]:

$$M_R=\sqrt{{M_{xT}}^2+{M_{yT}}^2}.$$

(2)

where: M_x and M_y are bending moments at the base of the column and are separated by an eccentricity from the center of the footing to the center of the column where the load and moments are applied as show in Figure 2.

Substituting M_xT = M_x + Pe_y and M_yT = M_y + Pe_x into Equation (2) is obtained as follows:

$$M_R=\sqrt{{\left(M_x+P{e}_y\right)}^2+{\left(M_y+P{e}_x\right)}^2}.$$

(3)

The angle of inclination where the maximum and minimum pressure section appears is:

$$\theta =arc sin\left[\frac{M_x+P{e}_y}{\sqrt{{\left(M_x+P{e}_y\right)}^2+{\left(M_y+P{e}_x\right)}^2}}\right].$$

(4)

Substituting A = πD²/4, I = πD⁴/64, M_R for an only moment into Equation (1), the maximum and minimum pressures of the circular footing are obtained as follows:

$${\sigma }_1=\frac{4P}{\pi D^2}+\frac{32\sqrt{{\left(M_x+P{e}_y\right)}^2+{\left(M_y+P{e}_x\right)}^2}}{\pi D^3},$$

(5)

$${\sigma }_2=\frac{4P}{\pi D^2}-\frac{32\sqrt{{\left(M_x+P{e}_y\right)}^2+{\left(M_y+P{e}_x\right)}^2}}{\pi D^3},$$

(6)

where σ₁ is the maximum pressure, σ₂ = minimum pressure, D is the diameter of the base.

2.1. Minimum Surface for Circular Isolated Footing

The minimum area A_min (objective function) is obtained as follows [16]:

$$A_{min}=\frac{\pi D^2}{4},$$

(7)

The constraint functions are shown in Equations (5) and (6), 0 ≤ σ₁, σ₂ ≤ σ_aasp (available admissible soil pressure). Available admissible soil pressure = Admissible soil pressure minus the concrete weight and soil fill weight.

2.2. Minimum Cost for Circular Isolated Footing

When substituting A = πD²/4, I_x = πD⁴/64, I_y = πD⁴/64, M_xT = M_ux + P_ue_y, M_yT = M_uy + P_ue_x into Equation (1) to find the factored pressure σ_u in terms of the coordinates for a circular footing, the general equation is as follows:

$${\sigma }_u\left(x, y\right)=\frac{4P_u}{\pi D^2}+\frac{64\left(M_{ux}+P_u{e}_y\right)y}{\pi D^4}+\frac{64\left(M_{uy}+P_u{e}_x\right)x}{\pi D^4},$$

(8)

where: P_u (Factored vertical axial load in kN) = 1.2P_D (Dead load) + 1.6P_L (Live load); M_ux (Factored moment on the X axis in kN-m) = 1.2M_xD (Dead load moment on the X axis) + 1.6M_xL (Live load moment on the X axis); M_uy = (Factored moment on the Y axis in kN-m) = 1.2M_yD (Dead load moment on the Y axis) + 1.6M_yL (Live load moment on the Y axis). These factors are according to the (ACI 318-14) [31].

The design of isolated footings is carried out by moments, bending shear and punching shear according to the (ACI 318-14) [31].

2.2.1. Moments

Figure 3 presents the critical moment sections for a circular footing under biaxial bending, assuming that the column is placed anywhere on the base (ACI 318-14) [31].

Below, the moments are developed with respect to the main axes (X and Y axes), since the reinforcing steel must be presented in these directions because, in a building, they must be drawn on the ground according to the main axes. If it was developed by the resultant moment, it would not be practical because, in each footing, it would be necessary to show the direction of the axis where the resultant moment appears.

Moment in the a axis “M_ua” is as follows:

$$M_{ua}=-{\int }_{e_y+\frac{c_y}{2}}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4y^2}}{2}}^{\frac{\sqrt{D^2-4y^2}}{2}}{\sigma }_u\left(x, y\right)\left(y-e_y-\frac{c_y}{2}\right)dxdy,$$

(9)

$$\begin{array}{ll}{M}_{ua}=& -\frac{\left[4\left(P_u{e}_y+M_{ux}\right){\left(c_y+2e_y\right)}^3+P_u{D}^2\left(2D^2+{c_y}^2-6e_y{c}_y-16{e_y}^2\right)\right]\sqrt{D^2-{\left(c_y+2e_y\right)}^2}}{6\pi D^4}\\ & +\frac{5M_{ux}\left(c_y+2e_y\right)\sqrt{D^2-{\left(c_y+2e_y\right)}^2}}{3\pi D^2}+\frac{\left(P_u{c}_y-2M_{ux}\right)}{4\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_y+2e_y}{D}\right)\right],\end{array}$$

(10)

where: c_y is the side in the Y direction of the column.

Moment in the b axis “M_ub” is as follows:

$$M_{ub}=\frac{P_u{c}_y}{2}+M_{ux}-{\int }_{e_y-\frac{c_y}{2}}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4y^2}}{2}}^{\frac{\sqrt{D^2-4y^2}}{2}}{\sigma }_u\left(x, y\right)\left(y-e_y+\frac{c_y}{2}\right)dxdy,$$

(11)

$$\begin{array}{ll}{M}_{ub}=& \frac{\left[4\left(P_u{e}_y+M_{ux}\right){\left(c_y-2e_y\right)}^3-P_u{D}^2\left(2D^2+{c_y}^2+6e_y{c}_y-16{e_y}^2\right)\right]\sqrt{D^2-{\left(c_y-2e_y\right)}^2}}{6\pi D^4}\\ & -\frac{5M_{ux}\left(c_y-2e_y\right)\sqrt{D^2-{\left(c_y-2e_y\right)}^2}}{3\pi D^2}+\frac{\left(P_u{c}_y+2M_{ux}\right)}{4\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_y-2e_y}{D}\right)\right].\end{array}$$

(12)

Moment in the c axis “M_uc” is:

$$M_{uc}=-{\int }_{e_x+\frac{c_x}{2}}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4x^2}}{2}}^{\frac{\sqrt{D^2-4x^2}}{2}}{\sigma }_u\left(x, y\right)\left(x-e_x-\frac{c_x}{2}\right)dydx,$$

(13)

$$\begin{array}{ll}{M}_{uc}=& -\frac{\left[4\left(P_u{e}_x+M_{uy}\right){\left(c_x+2e_x\right)}^3+P_u{D}^2\left(2D^2+{c_x}^2-6e_x{c}_x-16{e_x}^2\right)\right]\sqrt{D^2-{\left(c_x+2e_x\right)}^2}}{6\pi D^4}\\ & +\frac{5M_{uy}\left(c_x+2e_x\right)\sqrt{D^2-{\left(c_x+2e_x\right)}^2}}{3\pi D^2}+\frac{\left(P_u{c}_x-2M_{uy}\right)}{4\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_x+2e_x}{D}\right)\right],\end{array}$$

(14)

where: c_x is the side in the X direction of the column.

Moment in the e axis “M_ue” is as follows:

$$M_{ue}=\frac{P_u{c}_x}{2}+M_{uy}-{\int }_{e_x-\frac{c_x}{2}}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4x^2}}{2}}^{\frac{\sqrt{D^2-4x^2}}{2}}{\sigma }_u\left(x, y\right)\left(x-e_x+\frac{c_x}{2}\right)dydx,$$

(15)

$$\begin{array}{ll}{M}_{ue}=& \frac{\left[4\left(P_u{e}_x+M_{uy}\right){\left(c_x-2e_x\right)}^3-P_u{D}^2\left(2D^2+{c_x}^2+6e_x{c}_x-16{e_x}^2\right)\right]\sqrt{D^2-{\left(c_x-2e_x\right)}^2}}{6\pi D^4}\\ & -\frac{5M_{uy}\left(c_x-2e_x\right)\sqrt{D^2-{\left(c_x-2e_x\right)}^2}}{3\pi D^2}+\frac{\left(P_u{c}_x+2M_{uy}\right)}{4\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_x-2e_x}{D}\right)\right].\end{array}$$

(16)

2.2.2. One-Way Shear or Shear

Figure 4 presents the critical shear sections for a circular footing under biaxial bending, assuming that the column is placed anywhere on the footing. Critical shear sections appear at a distance d (effective footing depth) from the column faces (ACI 318-14) [31].

Shear in the f axis “V_ubf” is as follows:

$$V_{ubf}=-{\int }_{e_y+\frac{c_y}{2}+d}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4y^2}}{2}}^{\frac{\sqrt{D^2-4y^2}}{2}}{\sigma }_u\left(x, y\right)dxdy,$$

(17)

$$\begin{array}{ll}{V}_{ubf}=& \frac{\left\{3P_u{D}^2\left(c_y+2d+2e_y\right)-16\left(P_u{e}_y+M_{ux}\right)\left[D^2-{\left(c_y+2d+2e_y\right)}^2\right]\right\}\sqrt{D^2-{\left(c_y+2d+2e_y\right)}^2}}{3\pi D^4}\\ & -\frac{P_u}{2\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_y+2d+2e_y}{D}\right)\right].\end{array}$$

(18)

Shear in the g axis “V_ubg” is as follows:

$$V_{ubg}=P_u-{\int }_{e_y-\frac{c_y}{2}-d}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4y^2}}{2}}^{\frac{\sqrt{D^2-4y^2}}{2}}{\sigma }_u\left(x, y\right)dxdy,$$

(19)

$$\begin{array}{ll}{V}_{ubg}=& -\frac{\left\{3P_u{D}^2\left(c_y+2d-2e_y\right)+16\left(P_u{e}_y+M_{ux}\right)\left[D^2-{\left(c_y+2d-2e_y\right)}^2\right]\right\}\sqrt{D^2-{\left(c_y+2d-2e_y\right)}^2}}{3\pi D^4}\\ & +\frac{P_u}{2\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_y+2d-2e_y}{D}\right)\right].\end{array}$$

(20)

Shear in the h axis “V_ubh” is as follows:

$$V_{ubh}=-{\int }_{e_x+\frac{c_x}{2}+d}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4x^2}}{2}}^{\frac{\sqrt{D^2-4x^2}}{2}}{\sigma }_u\left(x, y\right)dydx,$$

(21)

$$\begin{array}{ll}{V}_{ubh}=& \frac{\left\{3P_u{D}^2\left(c_x+2d+2e_x\right)-16\left(P_u{e}_x+M_{uy}\right)\left[D^2-{\left(c_x+2d+2e_x\right)}^2\right]\right\}\sqrt{D^2-{\left(c_x+2d+2e_x\right)}^2}}{3\pi D^4}\\ & -\frac{P_u}{2\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_x+2d+2e_x}{D}\right)\right].\end{array}$$

(22)

Shear in the i axis “V_ubi” is as follows:

$$V_{ubi}=P_u-{\int }_{e_x-\frac{c_x}{2}-d}^{\frac{D}{2}}{\int }_{-\frac{\sqrt{D^2-4x^2}}{2}}^{\frac{\sqrt{D^2-4x^2}}{2}}{\sigma }_u\left(x, y\right)dydx,$$

(23)

$$\begin{array}{ll}{V}_{ubi}=& -\frac{\left\{3P_u{D}^2\left(c_x+2d-2e_x\right)+16\left(P_u{e}_x+M_{uy}\right)\left[D^2-{\left(c_x+2d-2e_x\right)}^2\right]\right\}\sqrt{D^2-{\left(c_x+2d-2e_x\right)}^2}}{3\pi D^4}\\ & +\frac{P_u}{2\pi }\left[\pi -2\mathit{arcsin}\left(\frac{c_x+2d-2e_x}{D}\right)\right].\end{array}$$

(24)

2.2.3. Two-Way Shear or Punching

Figure 5 presents the critical punching section for a circular footing under biaxial bending, assuming that the column is placed anywhere on the footing. The critical punching section appears at a distance d/2 from the four column faces (ACI 318-14) [31].

Punching shear “V_up” is as follows:

$$V_{up}=P_u-{\int }_{y_2}^{y_1}{\int }_{x_2}^{x_1}{\sigma }_u\left(x, y\right)dxdy,$$

(25)

$$V_{up}=-\frac{\left\{P_u\pi D^4-4\left[P_u{D}^2+8\left(P_u{e}_y+M_{ux}\right)\left(y_1+y_2\right)+8\left(P_u{e}_x+M_{uy}\right)\left(x_1+x_2\right)\right]\left(x_1-x_2\right)\left(y_1-y_2\right)\right\}}{\pi D^4}.$$

(26)

where x₁, y₁, x₂ and y₂ are the coordinates at each vertex of the critical section in both directions.

2.2.4. Objective Function

The C_min (minimum cost in dollars) is equal to the sum of the C_s (steel cost) and the C_c (concrete cost), these costs include manpower and materials. The equation is as follows:

$$C_{min}=V_c{C}_c+{V_s\gamma }_s{C}_s,$$

(27)

where: γ_s (Steel density) = 78 kN/m³.

The amount of steel is determined based on the reinforcement of the footing. The footing is reinforced in the form of a rectangular or square grid and a circular ring at a distance r (concrete cover in m) from the outer edge.

The reinforcing areas A_sx and A_sy in the X and Y directions are expressed as:

$$A_{sy}={\rho }_y{b}_{wx}d;A_{sx}={\rho }_x{b}_{wy}d,$$

(28)

where A_sy is the largest area of A_sya and A_syb, A_sx is the largest area of A_sxc and A_sxe, and these areas are obtained as follows:

$$A_{sya}={\rho }_x{b}_{wxa}d;A_{syb}={\rho }_x{b}_{wxb}d; A_{sxc}={\rho }_y{b}_{wyc}d; A_{sxe}={\rho }_y{b}_{wye}d,$$

(29)

where b_wxa, b_wxb, b_wyc and b_wyd are obtained as follows:

$$\begin{gathered} b_{wxa}=\sqrt{D^2-{\left(2e_y+c_y\right)}^2};b_{wxb}=\sqrt{D^2-{\left(2e_y-c_y\right)}^2}; \\ {b}_{wyc}=\sqrt{D^2-{\left(2e_x+c_x\right)}^2};b_{wye}=\sqrt{D^2-{\left(2e_x-c_x\right)}^2} , \end{gathered}$$

(30)

The Number of rods n_x and n_y in the two directions are given by:

$$n_x=\frac{A_{sx}}{a_s}; n_y=\frac{A_{sy}}{a_s},$$

(31)

where a_s is the cross-sectional area of the rod used and is considered equal in the two directions:

$$s_x=\frac{b_{wy}{a}_s}{A_{sx}}; s_y=\frac{b_{wx}{a}_s}{A_{sy}},$$

(32)

where s_x is the spacing of the reinforcing steel bars in the X direction, s_y is the spacing of the reinforcing steel bars in the Y direction.

The total length of the steel bars in the two directions is given by:

$$L_x=D+2\sum _{j=1}^{\left(n_x-3\right)/2}\sqrt{D^2-{\left(2j{s}_x\right)}^2},$$

(33)

$$L_y=D+2\sum _{i=1}^{\left(n_y-3\right)/2}\sqrt{D^2-{\left(2i{s}_y\right)}^2}.$$

(34)

Note: Equations (33) and (34) were developed from the bars located in the main axes (X and Y axes) and then the following expression is the length that decreases according to the separation of the bars. For bars parallel to the X axis, the length is $2\sqrt{{\left(\frac{D}{2}\right)}^2-{\left(j{s}_x\right)}^2}=\sqrt{D^2-{\left(2j{s}_x\right)}^2}$ (Length of the bars located on the positive part of the Y axis; these lengths must be multiplied by two to obtain the total length of the bars located outside the X axis.). For bars parallel to the Y axis, the length is $2\sqrt{{\left(\frac{D}{2}\right)}^2-{\left(i{s}_y\right)}^2}=\sqrt{D^2-{\left(2i{s}_y\right)}^2}$ (Length of the bars located on the positive part of the X axis; these lengths must be multiplied by two to obtain the total length of the bars located outside the Y axis.).

Length of the circumferential steel bar “L_c” is as follows:

$$L_c=\pi \left(D-2r\right),$$

(35)

where r is the concrete cover.

Figure 6 presents the A_sx, A_sy, s_x and s_y to show the lengths L_x, L_y and L_c.

The total volume of reinforcing steel “V_s” is as follows:

$$V_s=a_s\left(L_y+L_x+L_c\right).$$

(36)

The total volume of concrete “V_c” is as follows:

$$V_c=0.25\pi D^2\left(d+r\right)-a_s\left(L_y+L_x+L_c\right).$$

(37)

Substituting Equations (36) and (37) into Equation (27) is obtained as follows:

$$C_{min}=\left[0.25\pi D^2\left(d+r\right)-a_s\left(L_y+L_x+L_c\right)\right]C_c+\left[a_s\left(L_y+L_x+L_c\right)\right]{\gamma }_s{C}_s.$$

(38)

Now, substituting α = γ_sC_s/C_c → γ_sC_s = αC_c into Equation (38) is obtained as follows:

$$C_{min}=\left[0.25\pi D^2\left(d+r\right)+a_s\left(\alpha -1\right)\left(L_y+L_x+L_c\right)\right]C_c.$$

(39)

Substituting Equations (32)–(34) into Equation (38) is obtained as follows:

$$C_{min}=\left[0.25\pi D^2\left(d+r\right)+a_s\left(\alpha -1\right)\left(2D+2\sum _{j=1}^{\left(n_x-3\right)/2}\sqrt{D^2-{\left(2j{s}_x\right)}^2}+2\sum _{i=1}^{\left(n_y-3\right)/2}\sqrt{D^2-{\left(2i{s}_y\right)}^2}+\pi \left(D-2r\right)\right)\right]C_c.$$

(40)

2.2.5. Constraint Functions

For the design of circular footings, the equations according to the standard code of the American Concrete Institute (ACI 318-14) [31] are used.

For the moments, these are as follows:

$$\left|M_{ua}\right|\le {\mathrm{Ø}}_f{f}_{y}d{A}_{sya}\left(1-\frac{A_{sya}{f}_y}{1.7b_{wxa}d{f^{\prime }}_c}\right),$$

(41)

$$\left|M_{ub}\right|\le {\mathrm{Ø}}_f{f}_{y}d{A}_{syb}\left(1-\frac{A_{syb}{f}_y}{1.7b_{wxb}d{f^{\prime }}_c}\right),$$

(42)

$$\left|M_{uc}\right|\le {\mathrm{Ø}}_f{f}_{y}d{A}_{sxc}\left(1-\frac{A_{sxc}{f}_y}{1.7b_{wyc}d{f^{\prime }}_c}\right),$$

(43)

$$\left|M_{ue}\right|\le {\mathrm{Ø}}_f{f}_{y}d{A}_{sxe}\left(1-\frac{A_{sxe}{f}_y}{1.7b_{wyd}d{f^{\prime }}_c}\right),$$

(44)

where f_y is the specified yield strength of reinforcement of steel in MPa, f′_c is the specified compressive strength of the concrete at 28 days in MPa, A_sx and A_sy are the steel areas in the two directions, and Ø_f = 0.90 (bending strength reduction factor).

For the shears, the equations are [31] as follows:

$$\left|V_{ubf}\right|\le 0.17{\varnothing }_v\sqrt{{f^{\prime }}_c}{b}_{wxsf}d,$$

(45)

$$\left|V_{ubg}\right|\le 0.17{\varnothing }_v\sqrt{{f^{\prime }}_c}{b}_{wxsg}d,$$

(46)

$$\left|V_{ubh}\right|\le 0.17{\varnothing }_v\sqrt{{f^{\prime }}_c}{b}_{wysh}d,$$

(47)

$$\left|V_{ubi}\right|\le 0.17{\varnothing }_v\sqrt{{f^{\prime }}_c}{b}_{wysi}d,$$

(48)

$$\begin{gathered} b_{wxsf}=\sqrt{D^2-{\left(2e_y+c_y+2d\right)}^2};b_{wxsg}=\sqrt{D^2-{\left(2e_y-c_y-2d\right)}^2}; \\ {b}_{wysh}=\sqrt{D^2-{\left(2e_x+c_x+2d\right)}^2};b_{wysi}=\sqrt{D^2-{\left(2e_x-c_x-2d\right)}^2} , \end{gathered}$$

(49)

where Ø_v = 0.85 (shear strength reduction factor).

For the punching, the equation is [31] as follows:

$$\left|V_{up}\right|\le \left\{\begin{array}{c}0.17{\varnothing }_v\left(1+\frac{2}{{\beta }_c}\right)\sqrt{{f\prime }_c}{b}_{0}d\\ 0.083{\varnothing }_v\left(\frac{{\alpha }_{s}d}{b_0}+2\right)\sqrt{{f\prime }_c}{b}_{0}d\\ 0.33{\varnothing }_v\sqrt{{f\prime }_c}{b}_{0}d\end{array}\right.,$$

(50)

where β_c = long side of column/short side of column; b₀ is the critical punching perimeter in m; α_s is 40 for interior column; α_s is 30 for edge column; and α_s is 20 for corner column.

For the percentages of steel, they are [31] as follows:

$${\rho }_x,{\rho }_y\le 0.75\left[\frac{0.85{\beta }_1{f\prime }_c}{f_y}\left(\frac{600}{600+f_y}\right)\right],$$

(51)

$${\rho }_x,{\rho }_y\ge \left\{\begin{array}{c}\frac{0.25\sqrt{{f^{\prime }}_c}}{f_y}\\ \frac{1.4}{f_y}\end{array}\right.,$$

(52)

$$0.65\le {\beta }_1=\left(1.05-\frac{{f^{\prime }}_c}{140}\right)\le 0.85,$$

(53)

where β₁ is the factor relating the depth of the equivalent rectangular compressive stress block to the depth of the neutral axis.

For the steel areas, the equations are [31] as follows:

$$A_{sx}\ge \left\{\begin{array}{c}{A}_{sxc}={\rho }_x{b}_{wyc}d\\ A_{sxe}={\rho }_x{b}_{wye}d\end{array}\right.,$$

(54)

$$A_{sy}\ge \left\{\begin{array}{c}{A}_{sya}={\rho }_x{b}_{wxa}d\\ A_{syb}={\rho }_x{b}_{wxb}d\end{array}\right..$$

(55)

For the number of rods, they are [31] as follows:

$$n_x=\frac{A_{sx}}{a_s},$$

(56)

$$n_y=\frac{A_{sy}}{a_s}.$$

(57)

For the separations, they are [31] as follows:

$$s_x=\frac{b_{wy}{a}_s}{A_{sx}},$$

(58)

$$s_y=\frac{b_{wx}{a}_s}{A_{sy}},$$

(59)

The proposed model is shown in two phases. The first phase is to determine the smaller area, and the second phase is to obtain the minimum cost once the diameter of the footing is known. For the minimum surface (First phase), the independent variables are σ_max, P, M_x and M_y, and the dependent variables are A_min, D, σ₁ and σ₂. For the minimum cost (Second phase), the independent variables are D, P_u, M_ux and M_uy and the dependent variables are C_min, d, A_sx, A_sy, ρ_x and ρ_y.

The flowchart algorithm for the lowest cost design process of a circular footing is shown in Figure A1 (see Appendix A).

The flowchart for using Maple software (v.15) to design a circular footing with the lowest cost is presented in Figure A2 (see Appendix A).

3. Numerical Problems

Three numerical problems are shown (each example presents four variants) to find the lowest cost design of circular footings under biaxial bending. Problem 1: The center of the column is placed in the center of the footing. Problem 2: The center of the column is placed at a distance of 0.25 times the diameter of the footing from the center of the footing (e_x = D/4 and e_y = 0). Problem 3: One face of the column is placed on one of the footing faces (e_x = D/2 − c_x/2 and e_y = 0). The general data for the three problems are c_x = c_y = 50 cm, r = 7.5 cm, as = 5.07 cm² (1Ø1”), σ_aasp = 200 kN/m², f′_c = 21 MPa, f_y = 420 MPa and α = 90.

The data for Problem 1 with e_x = 0 and e_y = 0, and the input data (P, M_x and M_y) are shown in

Table 1. Input data for problem 1 (unfactored loads and moments).

Problem	PD (kN)	PL (kN)	P (kN)	MxD (kN-m)	MxL (kN-m)	Mx (kN-m)	MyD (kN-m)	MyL (kN-m)	My (kN-m)
1.A	800	700	1500	300	200	500	200	100	300
1.B	700	600	1300	300	200	500	200	100	300
1.C	600	500	1100	300	200	500	200	100	300
1.D	500	400	900	300	200	500	200	100	300

. The results are shown in

Table 2. Results of Problem 1.

Problem	D (m)	ex (m)	σ1 (kN/m2)	σ2 (kN/m2)	Amin (m2)	Pu (kN)	Mux (kN-m)	Muy (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
1.A	4.10	0	199.79	27.44	13.20	2080	680	400	64.39	64.39	32.05	32.05	47.50	0.00333	0.00333	10.85Cc
1.B	4.00	0	196.25	10.65	12.57	1800	680	400	56.19	63.55	35.82	31.67	42.50	0.00333	0.00377	9.73Cc
1.C	4.25	0	154.91	0.17	14.19	1520	680	400	59.72	63.89	35.82	33.49	42.50	0.00333	0.00356	10.79Cc
1.D	5.20	0	84.62	0.14	21.24	1240	680	400	64.74	64.69	40.57	40.60	37.50	0.00333	0.00333	14.12Cc

where: D, e_x and d are adjusted data.

.

The data for Problem 2 with e_x = D/4 and e_y = 0, and the input data (P, M_x and M_y) are shown in

Table 3. Input data for problem 2 (unfactored loads and moments).

Problem	PD (kN)	PL (kN)	P (kN)	MxD (kN-m)	MxL (kN-m)	Mx (kN-m)	MyD (kN-m)	MyL (kN-m)	My (kN-m)
2.A	800	600	1400	300	200	500	−500	−500	−1000
2.B	700	500	1200	300	200	500	−500	−500	−1000
2.C	600	400	1000	300	200	500	−500	−500	−1000
2.D	500	300	800	300	200	500	−500	−500	−1000

. The results are shown in

Table 4. Results of Problem 2.

Problem	D (m)	ex (m)	σ1 (kN/m2)	σ2 (kN/m2)	Amin (m2)	Pu (kN)	Mux (kN-m)	Muy (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
2.A	4.20	1.05	195.40	6.71	13.85	1920	680	−1400	86.79	86.79	24.36	24.36	62.50	0.00333	0.00333	15.12Cc
2.B	4.00	1.00	181.93	9.06	12.57	1640	680	−1400	71.35	69.41	28.21	29.00	52.50	0.00342	0.00333	11.60Cc
2.C	4.00	1.00	159.15	0	12.57	1360	680	−1400	82.63	56.19	24.36	35.82	42.50	0.00490	0.00333	10.33Cc
2.D	5.00	1.25	81.49	0	19.63	1080	680	−1400	97.48	62.07	25.85	40.60	37.50	0.00523	0.00333	14.62Cc

.

The data for Problem 3 with e_x = D/2 − c_x/2 and e_y = 0, and the input data (P, M_x and M_y) are shown in

Table 5. Input data for problem 3 (unfactored loads and moments).

Problem	PD (kN)	PL (kN)	P (kN)	MxD (kN-m)	MxL (kN-m)	Mx (kN-m)	MyD (kN-m)	MyL (kN-m)	My (kN-m)
3.A	500	400	900	100	100	200	−500	−500	−1000
3.B	400	300	700	100	100	200	−500	−500	−1000
3.C	300	200	500	100	100	200	−500	−500	−1000
3.D	200	100	300	100	100	200	−500	−500	−1000

. The results are shown in

Table 6. Results of Problem 3.

Problem	D (m)	ex (m)	σ1 (kN/m2)	σ2 (kN/m2)	Amin (m2)	Pu (kN)	Mux (kN-m)	Muy (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
3.A	3.30	1.40	198.20	12.25	8.55	1240	280	−1400	77.74	46.14	21.26	35.82	42.50	0.00561	0.00333	7.33Cc
3.B	3.00	1.25	188.01	10.05	7.07	960	280	−1400	64.88	51.77	23.13	28.99	52.50	0.00417	0.00333	6.63Cc
3.C	4.00	1.75	77.33	2.25	12.57	680	280	−1400	76.02	76.02	26.48	26.48	57.50	0.00333	0.00333	12.30Cc
3.D	6.40	2.95	18.29	0.36	32.17	400	280	−1400	94.24	90.32	34.32	35.82	42.50	0.00348	0.00333	24.52Cc

.

4. Results

The new model is verified as follows:

For moments:

• In Equation (10) substitute e_x = 0, e_y = D/2 − c_y/2 to obtain M_ua = 0.
• In Equation (12) substitute e_x = 0, e_y = − D/2 + c_y/2 to obtain M_ub = 0.
• In Equation (10) substitute _cy = 0, e_x = 0, e_y = 0 to obtain $M_{ua}=-\frac{P_{u}D}{3\pi }-\frac{M_{ux}}{2}$. In Equation (12) substitute c_y = 0, e_x = 0, e_y = 0 to obtain $M_{ub}=-\frac{P_{u}D}{3\pi }+\frac{M_{ux}}{2}$. Now, it is done M_ua − M_ub and it is given − M_ux. Therefore, the moments are in balance.
• In Equation (14) substitute e_x = D/2 − c_x/2, e_y = 0 to obtain M_uc = 0.
• In Equation (16) substitute e_x = − D/2 − c_x/2, e_y = 0 to obtain M_ue = 0.
• In Equation (14) substitute c_x = 0, e_x = 0, e_y = 0 to obtain $M_{uc}=-\frac{P_{u}D}{3\pi }-\frac{M_{uy}}{2}$. In Equation (16) substitute c_x = 0, e_x = 0, e_y = 0 to obtain $M_{ue}=-\frac{P_{u}D}{3\pi }+\frac{M_{uy}}{2}$. Now, it is done M_uc − M_ue and it is given − M_uy. Therefore, the moments are in balance.

For shears:

• In Equation (18) substitute e_x = 0, e_y = D/2 − c_y/2 − d to obtain V_ubf = 0.
• In Equation (20) substitute e_x = 0, e_y = − D/2 + c_y/2 + d to obtain V_ubg = 0.
• In Equation (18) substitute c_y = 0, d = 0, e_x = 0, e_y = 0 to obtain $V_{ubf}=-\frac{16M_{ux}}{3\pi D}-\frac{P_u}{2}$. In Equation (20) substitute c_y = 0, d = 0, e_x = 0, e_y = 0 to obtain $V_{ubg}=-\frac{16M_{ux}}{3\pi D}+\frac{P_u}{2}$. Now, it is done V_ubf − V_ubg and it is given − P_u. Therefore, the shears are in balance.
• In Equation (22) substitute e_x = D/2 − c_x/2 − d, e_y = 0 to obtain V_ubh = 0.
• In Equation (24) substitute e_x = −D/2 + c_x/2 + d, e_y = 0 to obtain V_ubi = 0.
• In Equation (22) substitute c_x = 0, d = 0, e_x = 0, e_y = 0 to obtain $V_{ubh}=-\frac{16M_{uy}}{3\pi D}-\frac{P_u}{2}$. In Equation (24) substitute c_x = 0, d = 0, e_x = 0, e_y = 0 to obtain $V_{ubi}=-\frac{16M_{uy}}{3\pi D}+\frac{P_u}{2}$. Now, it is done V_ubh − V_ubi and it is given − P_u. Therefore, the shears are in balance.

For punching:

• In Equation (26) substitute x₁ = c_x/2 + d/2, y₁ = c_y/2 + d/2, x₂ = −c_x/2 − d/2, y₂ = −c_y/2 − d/2 to obtain $V_{up}=-\frac{\left[P_u\pi D^2-4P_u\left(c_x+d\right)\left(c_y+d\right)\right]}{\pi D^2}$.

Table 1, Table 2 and Table 3 present the following: For Problems 1.A, 1.B, 2.A, 2.B, 3.A and 3.B, the maximum pressure governs, and for problems 1.C, 1.D, 2.C, 2.D, 3.C and 3.D, the minimum pressure governs.

Table 2 shows the following: when P_u is reduced, A_sx, A_sy, ρ_x and ρ_y grow, and A_min, s_x, s_y, d and C_min are reduced for the first two problems (Problems 1.A and I.B); when P_u is reduced, A_sx, A_sy, A_min, s_x, s_y and C_min grow, and ρ_y and d are reduced; ρ_x is the same for the last two problems (Problems 1.C and I.D).

Table 4 presents the following: when P_u is reduced, s_x and s_y grow, A_sx, A_sy, A_min, ρ_x, d and C_min are reduced; ρ_y is the same for the first two problems (Problems 2.A and 2.B); when P_u is reduced, A_sx, A_sy, A_min, s_x, s_y and C_min grow; ρ_x, ρ_y and d are reduced for the last two problems (Problems 2.C and 2.D).

Table 6 shows the following: when P_u is reduced, d, s_x and s_y grow; A_sx, A_sy, A_min, ρ_x, ρ_y and C_min are reduced for the first two problems (Problems 3.A and 3.B); when P_u is reduced, A_sx, A_sy, A_min, s_x, s_y, ρ_x, ρ_y and C_min grow; d is reduced for the last two problems (Problems 3.C and 3.D).

According to the results, it is observed that it can be applied to the footings located at the property boundary, as shown in Problem 3.

Therefore, this document is justified for the optimal cost design for circular footings if the column is located anywhere on the footing because, in the bibliographic review, only the optimal cost design for circular footings is presented when the center of the column coincides with the center of the footing.

Table 7. Comparison of the model proposed by Al-Ansari and Afzal (Example F1) against the new model.

Model	D (m)	ex (m)	σ1 (kN/m2)	σ2 (kN/m2)	Amin (m2)	Pu (kN)	Mux (kN-m)	Muy (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
Al-Ansari and Afzal	2.10	0	142.13	31.10	3.46	300	60	40	21.18	21.18	12.41	12.41	27.50	0.00372	0.00372	1.83Cc
New model	1.90	0	180.77	30.85	2.84	300	60	40	14.81	14.81	16.13	16.13	22.50	0.00350	0.00350	1.26Cc

shows the results of Example F1 derived by Al-Ansari and Afzal (2021) [21] for P_D = 200 kN, P_L = 100 kN, M_xD = 18 kN-m, M_xL = 24 kN-m, M_yD = 12 kN-m, M_yD = 16 kN-m, e_x = 0 m, e_y = 0 m, c_x = c_y = 0.30 m, r = 0.075 m, as = 1.27 cm² (1Ø1/2”), σ_max = 200 kN/m², σ_aasp = 181.50 kN/m², f′_c = 30 MPa, f_y = 400 MPa and α = 90. Here, the unfactored loads and moments are P = 300 kN, M_x = 42 kN-m and M_y = 28 kN-m.

Table 8. Comparison of the model proposed by Al-Ansari and Afzal (Example F7) against the new model.

Model	D (m)	ex (m)	σ1 (kN/m2)	σ2 (kN/m2)	Amin (m2)	Pu (kN)	Mux (kN-m)	Muy (kN-m)	Asx (cm2)	Asy (cm2)	sx (cm)	sy (cm)	d (cm)	ρx	ρy	Cmin (USD)
Al-Ansari and Afzal	6.40	0	168.44	142.41	32.17	6800	350	280	178.10	178.10	18.11	18.11	79.50	0.00358	0.00358	44.58Cc
New model	6.30	0	170.05	146.75	31.17	6800	350	280	169.81	169.81	18.69	18.69	77.50	0.00350	0.00350	42.01Cc

shows the results of Example F7 derived by Al-Ansari and Afzal (2021) [21] for P_D = 3000 kN, P_L = 2000 kN, M_xD = 185 kN-m, M_xL = 80 kN-m, M_yD = 120 kN-m, M_yD = 85 kN-m, e_x = 0 m, e_y = 0 m, c_x = c_y = 0.70 m, r = 0.075 m, as = 5.07 cm² (1Ø1”), σ_max = 200 kN/m², γ_g = 15 kN/m², γ_c = 25 kN/m², σ_aasp = 176.30 kN/m², f′_c = 30 MPa, f_y = 400 MPa and α = 90. Here, the unfactored loads and moments are P = 5000 kN, M_x = 265 kN-m and M_y = 205 kN-m.

Table 7 and Table 8 show the following: A_min, ρ_x, ρ_y, A_sx, A_sy, d and C_min are smaller, and s_x and s_y are greater in the new model compared to the model presented by Al-Ansari and Afzal. The new model shows a saving of 17.92% in the contact area with soil and of 31.15% in cost for Example F1 with respect to the model proposed by Al-Ansari and Afzal (Table 7). The new model shows a saving of 3.11% in the contact area with soil and of 5.76% in cost for Example F7 with respect to the model proposed by Al-Ansari and Afzal (Table 8).

The differences between the new model and the model proposed by Al-Ansari and Afzal are as follows: (1) The new model uses the minimum area; (2) The new model uses the minimum cost; (3) The ground pressure is linear.

5. Conclusions

In civil work, the substructure is the essential part of the structure since it is responsible for transmitting the loads of the columns or walls to the subsoil.

This research presents the design of circular footings under biaxial bending with minimal cost, assuming that the column is placed anywhere of the footing, supported on elastic soils, the footing is rigid, and the pressure diagram is linear.

The proposed model is shown in two phases. The first phase is to determine the smaller area, and the second phase is to obtain the minimum cost once the diameter of the footing is known. For the minimum surface (First phase), the independent variables are σ_max, P, M_x and M_y, and the dependent variables are A_min, D, σ₁ and σ₂. For the minimum cost (Second phase), the independent variables are D, P_u, M_ux and M_uy and the dependent variables are C_min, d, A_sx, A_sy, ρ_x and ρ_y.

The contributions of this work are as follows:

(1)

Several engineers use the trial-and-error practical to find the diameter of the footings under biaxial bending and the design is determined, taking into account the maximum and uniform pressure at the bottom of the footing;

(2)

Some works demonstrate the cost-effective design of circular footings under biaxial bending supported on elastic soils, but consider a column placed at the center of the footing;

(3)

The model can be used as a verification of the allowable load capacity of the soil, considering the objective function “σ_max”, and the constraint functions are the same for biaxial bending;

(4)

The moments, shears and punching are verified by equilibrium (see Section 4).

(5)

The proposed model can be applied to other building standards, taking into account the moment, shear and punching that must be resisted;

(6)

When the maximum pressure governs and P_u is reduced, A_min and C_min are reduced in the two first cases;

(7)

When the minimum pressure governs and P_u is reduced, A_min and C_min grow in the two last cases;

(8)

The new model shows a saving of 17.92% in the contact area with soil and of 31.15% in cost for Example F1 with respect to the model proposed by Al-Ansari and Afzal (Table 7);

(9)

The new model shows a saving of 3.11% in the contact area with soil and of 5.76% in cost for Example F7 with respect to the model proposed by Al-Ansari and Afzal (Table 8).

This work shows, in detail, the equations for moments, shear and punching, as well as optimization algorithms, which are its main advantage over other works.

Suggestions for future research include lower cost design of a circular isolated footing under biaxial bending due to a column placed anywhere on the footing, assuming that the contact surface of the base with the soil works partially to compression.